# A triangle has corners A, B, and C located at #(4 ,2 )#, #(2 ,6 )#, and #(8 ,4 )#, respectively. What are the endpoints and length of the altitude going through corner C?

Length of altitude

Now, the length of altitude CN is given by distance formula

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Length of the altitude

Let ,

Distance between two points

So ,

It is clear that ,

corner

So,

Length of the altitude

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The endpoints of the altitude going through corner C are the coordinates of C and the foot of the altitude, which is the point where the altitude intersects the side opposite corner C. The length of the altitude can be found using the distance formula between the endpoints.

To find the foot of the altitude, we need to find the equation of the line passing through point C that is perpendicular to the line containing the segment AB. Then, we find the intersection point of this perpendicular line with segment AB.

Let's denote the coordinates of point A as (x1, y1), point B as (x2, y2), and point C as (x3, y3).

Given: A (4, 2) B (2, 6) C (8, 4)

The equation of the line containing AB: [ y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1) ]

The slope of AB: [ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} ]

Perpendicular slope: [ m_{\perp} = -\frac{1}{m_{AB}} ]

Using point-slope form, the equation of the line perpendicular to AB passing through C: [ y - 4 = -\frac{1}{3}(x - 8) ]

Now, we solve this equation simultaneously with the equation of line AB to find the foot of the altitude.

After finding the foot of the altitude, we can calculate the length of the altitude using the distance formula between the endpoints (C and the foot of the altitude).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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